λ (lambda)

Definition

In the context of linear algebra, λ (lambda) represents an eigenvalue, which is a scalar that indicates how much a corresponding eigenvector is stretched or compressed during a linear transformation. When a matrix acts on an eigenvector, the result is a scalar multiple of that eigenvector, where the scalar is the eigenvalue. This relationship is fundamental in understanding systems of equations and stability in various fields including economics.

5 Must Know Facts For Your Next Test

1. The equation defining the relationship between a matrix A, an eigenvector v, and its eigenvalue λ is given by the equation A*v = λ*v.
2. Eigenvalues can be real or complex numbers, depending on the characteristics of the matrix.
3. The multiplicity of an eigenvalue refers to how many times it appears as a root of the characteristic polynomial.
4. To find the eigenvalues of a matrix, one typically solves the characteristic equation |A - λI| = 0, where I is the identity matrix.
5. Eigenvalues are essential in determining the stability of equilibrium points in economic models, as they can indicate whether perturbations will die out or grow over time.

Review Questions

• How do eigenvalues and eigenvectors interact in the context of linear transformations?
  • Eigenvalues and eigenvectors are closely related in that an eigenvector remains in its direction even when transformed by a linear transformation represented by a matrix. The corresponding eigenvalue indicates how much the eigenvector is stretched or compressed. This relationship reveals key insights into the behavior of systems modeled by matrices, especially in terms of stability and response to changes.
• What role does the characteristic polynomial play in finding eigenvalues for a given matrix?
  • The characteristic polynomial is crucial for determining the eigenvalues of a matrix because it establishes the relationship between the matrix and its eigenvalues. By setting the characteristic polynomial equal to zero, one can find all possible values of λ (lambda) that satisfy the equation. This process enables us to identify key properties of the matrix, such as stability and long-term behavior in economic models.
• Evaluate how the properties of eigenvalues can influence economic models in terms of stability and growth.
  • The properties of eigenvalues significantly influence economic models, particularly when assessing the stability of equilibrium points. If all eigenvalues associated with a system have negative real parts, perturbations will diminish over time, suggesting stability. Conversely, if any eigenvalue has a positive real part, it indicates potential growth or instability in response to shocks. This evaluation is essential for economists when modeling dynamic systems and predicting long-term outcomes.

Related terms

Eigenvector: An eigenvector is a non-zero vector that changes at most by a scalar factor when a linear transformation is applied to it.

Matrix: A matrix is a rectangular array of numbers arranged in rows and columns, often used to represent linear transformations and systems of equations.

Characteristic Polynomial: The characteristic polynomial is a polynomial equation derived from a matrix that helps in finding its eigenvalues by setting it equal to zero.